Create animation of packaging that closes with rolls of film

[opinion-intention]

I want to create an animation where I have a square sheet of paper wrapping around my square box, coming up to form a gift box, with the side folds following my folds. How can I do this? i've tried bend, a line extrusion controlled by points, and rigging, but I haven't come up with it.
I looked at this tutorial https://www.youtube.com/watch?v=3VPyRcIDtok but it was not helpful to me.

To explain further, I would like to recreate an animation of the procedure of wrapping a gift (with clingfilm) similar to this one.
https://www.youtube.com/watch?v=S5T4TdSDMdQ

You should create the rounded corners and the bend not rigid, but as if gently bent by the passage of a roller.